Re: how

Am 25.04.2024 um 21:23 schrieb Chris M. Thomasson:
 > what about:
 >
 > [0] = 0
 > [1] = 1/3
 > [2] = 2/3
 > [3] = 1
Yes. Now look at Cantor's approach (if you don't know it already):
[1] = 1/1
[2] = 1/2
[3] = 2/1
[4] = 1/3
[5] = 2/2
[6] = 3/1
[7] = 1/4
[8] = 2/3
[9] = 3/2
[10]= 4/1
  :
This shows that the (positive) fractions are "countable". (This implies that the rational numbers are "countable" or [maybe a better term] "enumerable".)
 > .(3) for 1/3 is just how base 10 represents 1/3. Think about hitting a cycle during any long division in base 10. We can stop the process at any detected "cycle"... Once we hit a cycle its a rational. Fair enough? Or did I fuck up again?
No. We can indeed prove that a real number is rational iff its decimal expansion contains such a cycle.